Variational Aspects of the Seiberg-Witten Functional 3

VARIATIONAL ASPECTS OF THE SEIBERG-WITTEN FUNCTIONAL Jurgen¨ Jost, Xiaowei Peng and Guofang Wang 0. Introduction Recently, Seiberg and Witten (see [SW1], [SW2] and [W]) introduced a new monopole equation which yields new differential-topological invariants of four di- mensional manifolds, closely related to the Donaldson polynomial invarints [DK]. This equation has been used to give more elementary proof of many heorems in gauge theory and to obtain many new results (see [KM], [Le], [T1], [T2] and [T4]). These equations are first order equations, but the solutions also satisfy more general second order equations, in a similar way as (anti) self dual connections are solutions of the Yang-Mills equations or holomorphic maps between Kähler manifolds are harmonic. Sometimes, one can use index theorems or methods from algebraic geometry to construct solutions or to get information about the space of solutions. Another strategy would be to first solve the second order equations and then try to identify conditions under which certain solutions of the second order equations actually also solve the first order ones. This strategy has been successful in many other instances. One of the reasons for this success usually was the variational structures of the second order equations, namely, one could identify them as the Euler-Lagrange equations of some variational integral. The solutions of the first order equations are absolute minimum of this functional. arXiv:dg-ga/9504003v1 28 Apr 1995 We believe that this strategy should also be usefully for the Seiberg-Witten equations. We therefore study the corresponding variational integral which we call Seiberg-Witten functional. We show that weak solutions of the Euler-Lagrange equations are smooth. The main result of our paper is that the functional satisfies the Palais-Smale condition. As a consquence, one not only obtains the existence of the minimizers (among which one hopes to find solutions of the first order equations), but also the existence of unstable solutions. More precisely, one knows how to deduce from the Palais-Smale conditon the mountain pass lemma and a general Ljusternik-Schnirelmann type theory, and even Morse (-Bott) theory if the functi- ional is a Morse (-Bott) functional. We hope that these additional non-minimizing solutions (which do not solve the first order equations) also carry useful geometric information and can be used to define more invariants for differential 4-manifolds. This will be studied elsewhere, however. When we had already nearly completed this paper, the questions that are an- swered here were also posed by P. Braam at a conference at the ICTP in Trieste. 1991 Mathematics Subject Classification. 58E15, 53C07. Key words and phrases. Seiberg-Witten equation, Seiberg-Witten functional, regularity and Palais-Smale condition. Typeset by AMS-TEX 1 2 JURGEN¨ JOST, XIAOWEI PENG AND GUOFANG WANG This paper was written while the third auther was a guest of the research project SFB 237 of the DFG at the Ruhr University Bochum. The first auther was also supported by the DFG. 1. The Seiberg-Witten equation In order to be able to write down the equation of Seiberg-Witten, we need to recall the definition of a Spinc-structure. (For details see [LM]). Let (X,g) be an oriented, compact Riemannian 4-manifold and PSO(4) → X c c its oriented orthonormal frame bundle. Spin (4) = Spin(4) ×Z2 U(1). A Spin - structure is a lift of the structure group SO(4) to Spinc(4), i.e. there exists a c principal Spin -bundle PSpinc(4) → X such that there is a bundle map PSpinc(4) −→ PSO(4) ↓ ↓ X −→ X Let Q = PSpinc(4)/Spin(4) be a principal U(1)-bundle. W = PSpinc(4) ×Spinc(4) 4 C and L = Q ×U(1) C resp. is the associated spinor bundle and the line bundle resp.. W can be decomposed globally as W + and W −. Locally, W ± = S± ⊗ L1/2. Here S± is a spinor bundle w. r. t. a local Spin-struture on X. Both S± and L1/1 are locally defined. There exists a Clifford multiplication TX × W + → W − denoted by e · φ ∈ W − for e ∈ TX and φ ∈ W +. Here TX is the tangent bundle of X. A connection on the bundle W + can be defined by the Levi-Civita connection + − and a connection on L. The “twisted” Dirac operator DA : Γ(W ) → Γ(W ) is defined by 4 DA = ei ·∇A. i=1 X ± ± Here, Γ(W ) is the space of sections of W , {ei} is an orthonormal basis of TX and + ∇A is a connection on W induced by the Levi-Civita connection and a connection A on the line bundle L. Definition 1.1. The Seiberg-Witten equations are DAφ =0, (1.1) i F + = he e φ, φiei ∧ ej , A 4 i j + for A a connection on L and φ ∈ Γ(W ), where F (A) = −iFA is the curvature of + i A, FA is the self dual part of FA and {e } is the dual basis of {ei}. VARIATIONAL ASPECTS OF THE SEIBERG-WITTEN FUNCTIONAL 3 Definition 1.2. The Seiberg-Witten functional is i (1.2) SW (A, φ) = (|D |2 + |F + − he e φ, φiei ∧ ej|2)dvol A A 4 i j ZX The Euler-Lagrange equations of the Seiberg-Witten functional are ∗ i + 1 DADAφ − FA · φ − heiej φ, φieiejφ =0, (1.3) 2 8 i 1 d∗(F + − he e φ, φiei ∧ ej) + ImhD φ, e φiei =0. A 4 i j 2 A i ∗ ∗ Here, DA is the formal adjoint operator of DA, d = −∗ d∗ and ∗ is the Hodge star operator. It is easy to see that a solution of (1.1) is a solution of (1.3). In fact, it is clear from the definition of the functional that a solution of (1.1) is a minimizer of the Seiberg-Witten equation. The following Weitzenböck formula plays an important role in the Seiberg-Witten theory, s i (1.4) D∗ D φ = −∆ φ + φ + F · φ, A A A 4 2 A where s is the scalar curvature of (X,g). By this formula, the Seiberg-Witten functional can be rewritten as s 1 (1.5) SW (A, φ) = (|∇ φ|2 + |F +|2 + |φ|2 + |φ|4)dvol. A A 4 8 ZX And the Euler-Lagrange equation (1.3) can be rewritten as s 1 2 −∆Aφ + φ + |φ| φ =0, (1.6) 4 4 1 d∗F + + Imh∇ φ, φiei =0. A 2 i Here ∆A is the analyst’s Laplacian, the negative Laplacian, and ∇i = ∇ei . Lemma 1.3. For a smooth solution (A, φ) of equation (1.3) (or (1.6)), |φ|(x) ≤ max{−s0, 0}, where s0 = min{s(x)|x ∈ X}. Proof. From the maximum principle. Corollary 1.4. If the scalar curvature of X is nonnegative, then for any smooth solution (A, φ) of (1.2), φ ≡ 0. Hence, a solution of (1.2) shares many properties with solutions of the Seiberg- Witten equation (1.1). In section 3 below, we shall prove Lemma 1.3 for weak solutions of (1.2). In this paper we shall consider the Seiberg-Witten functional and prove a com- pactness theorem. The precise set-up of the problem will be given in section 2. Before ending the present section, we want to give the definition of the Palais- Smale condition. 4 JURGEN¨ JOST, XIAOWEI PENG AND GUOFANG WANG Let M be a Banach manifold and f : M → R a smooth functional. Let G be a Lie group acting on M and suppose f is invariant under G, i.e. for any g ∈ G and x ∈ X, f(gx) = f(x). f is said to satisfy the Palais-Smale condition: if for any sequence {xi}i∈N with (i) f(xi) is bounded, (ii) df(xi) → 0, as i →∞, there exists a subsequence (also denoted by {xi}) and a sequence gi ∈ G such that gixi converges in X to a critical point x of f, i.e. df(x) = 0, with f(x) = limi→∞ f(xi). 2. The set-up We need to choose a suitable working space to discuss the Seiberg-Witten functional. For a vector bundle with a metric over X, we define Lk,p(E), the Sobolev space of sections of E by completing the space Γ(E) of smooth sections of E by k α p 1/p kskLk,p =( |∇ s| ) , |αX|=0 Z where ∇ is a fixed metric connection on E. (For details see [P]). For such Sobolev spaces, the Sobolev embedding theorem and the Hölder inequality are valid, i.e. 4 1/4 1,2 (2.1) ( |s| ) ≤ ckskL1.2 , for s ∈ L , ZX and 2 2 1/2 4 1/4 4 1/4 (2.2) ( |s1| |s2| ) ≤ ( |s1| ) ( |s2| ) . ZX ZX ZX Let L1,2(W +) be the space of sections of the bundle W + of class L1,2 defined above and A1,2 = L1,2(A) the space of connections of class L1,2 defined by a fixed 1,2 + 1,2 connection A0. L (W ) and A are Hilbert manifolds. Lemma 2.1. The Seiberg-Witten functional SW is well defined on A1,2×L1,2(W +) and SW is smooth. Proof. From (2.1) and (2.2), 2 |Aφ| ≤ ckAkL1,2 kφkL1,2 . ZX This implies that SW is well-defined. It is easy to check that SW is smooth. Now let us to choose a suitable Lie group as a gauge group. First let G0 = 2,2 exp(iL (X, R)). We claim that G0 is a Lie group. Actually, G0 can be seen as 2,2 a quotient of L (X, R) under an equivalence relation ∼.

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